170 INDUCTION. 



expressions varied by putting x for a and a for #, 

 wherever that substitution would alter the value : to 

 which perhaps we ought to add (with M. Comte) sin x, 

 and arc (sin = x) . All other functions of x are 

 formed by putting some one or more of the simple 

 functions in the place of x or a, and subjecting them 

 to the same elementary operations. 



In order to carry on general reasonings on the sub- 

 ject of Functions, we require a nomenclature enabling 

 us to express any two numbers by names which, with- 

 out specifying what particular numbers they are, shall 

 show what function each is of the other ; or, in other 

 words, shall put in evidence their mode of formation 

 from one another. The system of general language 

 called algebraical notation does this. The expressions 

 a and a 2 + 3 a denote, the one any number, the other 

 the number formed from it in a particular manner. 

 The expressions a, b, n, and (a + b) n , denote any 

 three numbers, and a fourth which is formed from 

 them in a certain mode. 



The following may be stated as the general pro- 

 blem of the algebraical calculus : F being a certain 

 function of a given number, to find what function F 

 will be of any function of that number. For example, 

 a binomial a + b is a function of its two parts a and b, 

 and the parts are, in their turn, functions of a + b : 

 now (a + b) n is a certain function of the binomial ; 

 what function will this be of a and 6, the two parts ? 

 The answer to this question is the binomial theorem. 



The formula (a + b) n = a n +-^ a n - ] b + a n ~ 2 6 2 +, 



1 1.2 



&c., shows in what manner the number which is 

 formed by multiplying a + b into itself n times, might 

 be formed without that process, directly from a, b, and 

 n. And of this nature are all the theorems of the 



